We have the following relations
\(\phi _a = r^3(\nabla _a \rho + e_{ab}\nabla ^b\sigma ),\,\,\, \underline{\phi }_a = r^3(-\nabla _a \rho + e_{ab}\nabla ^b\sigma ).\) 
Moreover \(r^2\rho \)  and \(r^2\sigma \)  satisfy the scalar Fackerell-Ipser equation, in precisely a scalar wave equation
\(\Box _g\psi = \frac{1}{F}L\underline{L}\psi - {\Delta }\psi =0.\) 
Since \(r{\nabla }_a = {\nabla }_{\partial _{\theta }^a}\) , we have the commutators \([r\nabla ,{\nabla }_L]= [r\nabla ,{\nabla }_{\underline{L}}]=0\) .
Applying the projected covariant angular derivative \(r{\nabla }\)  on the unit 2-sphere \(\mathbb {S}^2\)  to the scalar wave equation (REF ), we get a new equation i.e the tensorial Fackerell-Ipser equation with the appearance of the additional zeroth order term since \([r{\nabla },{\Delta }]= \dfrac{1}{r^2}\) . Therefore, we can also obtain the vectorial Fackerell-Ipser equations (REF ) and (REF ) from the scalar wave equation (REF ) (see Remark 2.10 [1]} and also Remark 7.1 [2]}).
